The host's efforts paid off with 17 minutes played as Mainz' Leandro Barreiro fouled onrushing Dani Olmo inside the area. Forsberg broke the deadlock with a well-placed shot from the spot into the bottom right corner.
Dominic Dale, the world number 70, also reached Stage Two after topping his group with two wins and one draw.
It will be Favre's primary task to balance the exuberance of his exciting young starlets and experience of some of the older heads in the team. Enditem
"We had two big chances in the first half but Dortmund were more effective and clinical in front of the target today," Monchengladbach head coach Rose said.
Second seed Karolina Pliskova of the Czech Republic also met little trouble in her 6-4, 6-3 win over Russia's Anna Blinkova.
The 33-year-old Serb now faces a semifinal against Casper Ruud of Norway, who beat fourth seed Matteo Berrettini of Italy 4-6, 6-3, 7-6 (5).
Along with fellow new boy Allan, the Colombian was lively throughout and the Toffees fully deserved their win, with Dominic Calvert-Lewin's thumping header proving decisive.
Earlier in the day, women's 200m saw Kong Lingwei win with 23.64s as top favorite Ge failed to finish the race.
"This year is a special year. Playing with the Beijing Ducks for one year, I will never forget that I see these supporters every day, which makes me so touched. Thank you all. I love you," Lin noted.
To keep fit, the club is holding separate training sessions of not more than five players each at different locations while beefing up its squad in readiness for continental football and defending their crown.
"I didn't think I was totally ready for the event, but I was happy that I wasn't affected by others and finally made it," said the veteran Liu.